Error measures and solution artifacts of the harmonic balance method on the example of the softening Duffing oscillator

• Autorzy: Dänschel Hannes , Lentz Lukas , von Wagner Utz

Identyfikatory

DOI

10.15632/jtam-pl/186718

• Języki publikacji: EN

Abstrakty

EN

The Harmonic BalanceMethod (HBM) is one of the most often applied semi-analytic approximation methods in nonlinear dynamics. In earlier publications, the two coauthors already observed for the softening Duffing oscillator and other systems that especially low ansatz order HBM solutions may contain larger errors for some solution branches, and called this artifacts. In the present work, this problem is studied systematically with a new implementation of the method and applied again to the example of the softening Duffing oscillator. In conjunction with a mathematical definition for HBM artifacts we discuss and present possible a posteriori and a priori HBM error measures.

Słowa kluczowe

EN

Harmonic Balance Method; HBM; artifact solutions; softening Duffing oscillator

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2024
• Tom: Vol. 62 nr 2
• Strony: 435--455
• Opis fizyczny: Bibliogr. 28 poz., rys.

Twórcy

autor

Dänschel Hannes

• Technische Universität Berlin, Institut für Mechanik, Berlin, Germany

• https://orcid.org/0009-0009-3182-7784

autor

Lentz Lukas

• Hochschule Trier, Institut für Betriebs- und Technologiemanagement, Trier, Germany

• https://orcid.org/0009-0002-6091-3146

autor

von Wagner Utz

• Technische Universität Berlin, Institut für Mechanik, Berlin, Germany

• https://orcid.org/0000-0001-7914-3546

Bibliografia

• 1. ALT H., GODAU M., 1995, Computing the Fréchet distance between two polygonal curves, International Journal of Computational Geometry and Applications, 5, 75-91.
• 2. BASU S., POLLACK R., ROY M., 2006, Algorithms in Real Algebraic Geometry, 2nd ed., Springer Berlin, Heidelberg.
• 3. DE TERÁN F., DOPICO F.M., PÉREZ J., 2013, Condition numbers for inversion of Fiedler companion matrices, Linear Algebra and its Applications, 439, 4 944981.
• 4. DEUFLHARD P., 2011, Newton Methods for Nonlinear problems: Affine Invariance and Adaptive Algoirithms, Springer Series in Computational Mathematics, Springer Berlin, Heidelberg.
• 5. DEUFLHARD P., FIEDLER B., KUNKEL P., 1987, Efficient numerical pathfollowing beyond critical points, SIAM Journal on Numerical Analysis, 24, 4, 912-927.
• 6. DEUFLHARD P., HOHMANN A., 2019, Eine algorithmisch orientierte Einfuhrung, De Gruyter, Berlin, Boston.
• 7. DUFFING G., 1918, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Samlung Vieweg.
• 8. EDELMAN A., MURAKAMI H., 1995, Polynomial roots from companion matrix eigenvalues, Mathematics of Computation, 64, 763-776.
• 9. FERRI A.A., LEAMY M.J., 2009, Error estimates for harmonic-balance solutions of nonlinear dynamical systems, Collection of Technical Papers - AIAA/ ASME/ ASCE/ AHS/ ASC Structures, Structural Dynamics and Materials Conference, May, 115.
• 10. Figueira J.P., 2023, Discret-frechet, https://github.com/joaofig/discrete-frechet.
• 11. GARCÍA-SALDAÑA J.D., GASULL A., 2013, A theoretical basis for the Harmonic Balance Method, Journal of Differential Equations, 254, 1, 67-80.
• 12. HAGEDORN P., 1981, Non-linear Oscillations, Clarendon Press, Oxford and New York.
• 13. HERMAN R.L., 2016, An Introduction to Fourier Analysis, CRC Press.
• 14. HOLMES P.J., RAND D.A., 1976, The bifurcations of Duffing's equation: An application of catastrophe theory, Journal of Sound and Vibration, 44, 2, 237-253.
• 15. KOGELBAUER F., BREUNUNG T., 2021, When does the method of harmonic balance give a correct prediction for mechanical systems, Applicable Analysis, 102, 2, 425-443.
• 16. KOVACIC I., BRENNAN M.J., 2011, The Duffing Equation: Nonlinear Oscillators and their Phenomena, Wiley.
• 17. KRACK M., GROSS J., 2019, Harmonic Balance for Nonlinear Vibration Problems, Springer.
• 18. LENTZ L., VON WAGNER U., 2020, Avoidance of artifacts in harmonic balance solutions for nonlinear dynamical systems, Journal of Theoretical and Applied Mechanics, 58, 2, 307-316.
• 19. NAYFEH A.H., MOOK D.T., 1979, Nonlinear Oscillations, Wiley.
• 20. NOVAK S., FREHLICH R.G., 1982, Transition to chaos in the Duffing oscillator, Physical Review A,26, 6, 3660-3663.
• 21. STROGATZ S.H., 1994, Nonlinear Dynamics and Chaos, Westview Press.
• 22. UEDA Y., 1991, Survey of regular and chaotic phenomena in the forced Duffing oscillator, Chaos, Solitons and Fractals, 1, 3, 199-231.
• 23. URABE M., 1965, Galerkin's procedure for nonlinear periodic systems, Archive for Rational Mechanics and Analysis, 20, 120-152.
• 24. VON WAGNER U., LENTZ L., 2016, On some aspects of the dynamic behavior of the softening Duffing oscillator under harmonic excitation, Archive of Applied Mechanics, 86, 1383-1390.
• 25. VON WAGNER U., LENTZ L., 2018, On artifact solutions of semi-analytic methods in nonlinear dynamics, Archive of Applied Mechanics, 88, 1713-1724.
• 26. VON WAGNER U., LENTZ L., 2019, On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators, Applied Mathematical Modelling, 65, 408-414.
• 27. WOIWODE L., BALAJI N.N., KAPPAUF J., TUBITA F., GUILLOT L., et al., 2020, Comparison of ANM and predictor-corrector method to continue solutions of harmonic balance equations, [In:] Conference Proceedings of the Society for Experimental Mechanics Series.
• 28. WOIWODE L., KRACK M., 2023, Are Chebyshev-based stability analysis and Urabe's error bound useful features for Harmonic Balance?, Mechanical Systems and Signal Processing, 194, 110265.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).